General parameter-shift rules for quantum gradients

TL;DR

This paper introduces new general parameter-shift rules for quantum gradients in variational quantum algorithms, enabling more efficient derivative calculations and extending existing optimization methods.

Contribution

It derives closed-form, general parameter-shift rules for single- and multi-parameter quantum gates, improving resource efficiency for higher-order derivatives.

Findings

Reduces circuit evaluations for derivatives in quantum algorithms

Provides closed-form expressions for parameter-shift rules

Extends optimization algorithms with higher-order derivative reconstructions

Abstract

Variational quantum algorithms are ubiquitous in applications of noisy intermediate-scale quantum computers. Due to the structure of conventional parametrized quantum gates, the evaluated functions typically are finite Fourier series of the input parameters. In this work, we use this fact to derive new, general parameter-shift rules for single-parameter gates, and provide closed-form expressions to apply them. These rules are then extended to multi-parameter quantum gates by combining them with the stochastic parameter-shift rule. We perform a systematic analysis of quantum resource requirements for each rule, and show that a reduction in resources is possible for higher-order derivatives. Using the example of the quantum approximate optimization algorithm, we show that the generalized parameter-shift rule can reduce the number of circuit evaluations significantly when computing derivatives with respect to parameters that feed into many gates. Our approach additionally reproduces reconstructions of the evaluated function up to a chosen order, leading to known generalizations of the Rotosolve optimizer and new extensions of the quantum analytic descent optimization algorithm.